# Lecture series: Local positivity in Algebraic Geometry and Symplectic Topology

### Dr. Thomas Eckl (Liverpool)

## Abstracts of Lectures

In the week of 15.01.2018, Dr. Thomas Eckl from Liverpool, UK will give three lectures on the topic "Local positivity in Algebraic Geometry and Symplectic Topology". The first two lectures will be introductory and face to both undergraduate and graduate students interested in this topic; the third lecture is on the research level. For questions please contact Dr. Xin Fang (xfang@math.uni-koeln.de).Please find below the abstracts of the three lectures.

Lecture 1: Local Positivity in Algebraic Geometry and Symplectic Topology

After presenting interpolation problems in the complex plane and formulating the Segre-Harbourne-Gimigliano-Hirschowitz (SHGH) conjecture and Nagata's conjecture in this context, we introduce Seshadri constants as an algebraic-geometric measure of local positivity. We restate Nagata's conjecture in terms of Seshadri constants, and calculate them on del Pezzo surfaces. Then we discuss lower bounds to Seshadri constants. Finally, we show how Seshadri constants translate to Kaehler geometry, and what this tells us on symplectic packing problems.

Lecture 2: Newton-Okounkov bodies and moment maps

We construct Newton-Okounkov (N-O-) bodies of graded linear systems on algebraic varieties and discuss some of the algebraic-geometric invariants encoded in them. In particular, we show how their form determines Seshadri constants on the projective complex plane. Connecting N-O-bodies to toric moment polytopes we ask whether we can find an analogue of the toric moment map for general N-O-bodies, and discuss examples of such moment maps following a construction by Harada and Kaveh.

Lecture 3: Seshadri constants and Kaehler packings

We introduce Kaehler packing constants on projective algebraic varieties and show that these constants are equal to the corresponding Seshadri constants. We also discuss how this equality gives rise to possibly new explicit symplectic packings of the projective complex plane.

## Time and location:

Lecture 1: 15.01.2018, 14h15-15h45, Cohen-Vossen Raum 3.13.Lecture 2: 16.01.2018, 14h-15h30, Cohen-Vossen Raum 3.13.

Lecture 3: 17.01.2018, 14h-15h, Seminarraum 1, 0.05.

## Support:

This lecture is under the framework of the Graduiertenschule Mathematik/Informatik Köln, supported by the Graduiertenschule and the SFB TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics". Arbeitsgruppe Algebra und Zahlentheorie, Mathematisches Institut, Universität zu Köln